Characters andK-theory of discrete groups

Discrete Series Characters as Lifts from Two-structure Groups

Let G be a connected reductive Lie group with a relatively compact Cartan subgroup. Then it has relative discrete series representations. The main result of this paper is a formula expressing relative discrete series characters on G as “lifts” of relative discrete series characters on smaller groups called two-structure groups for G. The two-structure groups are connected reductive Lie groups w...

Group Theory: An Application of Discrete Groups

In order to analyze energy levels, bonding, and spectroscopy, it is of great importance to be able to determine the symmetry of a molecule. The symmetry of a molecule or atom can be analyzed strictly from the molecule itself, or from the environment of the atom. Symmetry properties are classified according to different groups determined by sets of symmetry operations. Different elements are con...

Structural Properties of Inner Amenable Discrete Groups

Hyperbolic Manifolds, Discrete Groups and Ergodic Theory

1 Ergodic theory References for this section: CFS]. 1. The basic setting of ergodic theory: a measure-preserving transformation T of a probability space (X; B; m). Usually we assume T is invertible. (More generally, measure-preserving means R f T = R f; equivalently, m(T ?1 (A)) = mA.) How many measure spaces are there? Standard Borel spaces: any Borel subset of a complete, separable metric spa...

Discrete Morse Theory and Graph Braid Groups

If Γ is any finite graph, then the unlabelled configuration space of n points on Γ, denoted UCΓ, is the space of n-element subsets of Γ. The braid group of Γ on n strands is the fundamental group of UC Γ. We apply a discrete version of Morse theory to these UCΓ, for any n and any Γ, and provide a clear description of the critical cells in every case. As a result, we can calculate a presentation...